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This research introduces a sophisticated mathematical model for understanding the transmission dynamics of COVID-19, incorporating both integer and fractional derivatives. The model undergoes a rigorous analysis, examining equilibrium points, the reproduction number, and feasibility. The application of fixed point theory establishes the existence of a unique solution, demonstrating stability in the model. To derive approximate solutions, the generalized Adams-Bashforth-Moulton method is employed, further enhancing the study's analytical depth. Through a numerical simulation based on Thailand's data, the research delves into the intricacies of COVID-19 transmission, encompassing thorough data analysis and parameter estimation. The study advocates for a holistic approach, recommending a combined strategy of precautionary measures and home remedies, showcasing their substantial impact on pandemic mitigation. This comprehensive investigation significantly contributes to the broader understanding and effective management of the COVID-19 crisis, providing valuable insights for shaping public health strategies and guiding individual actions.

In this article, we deal with mild solution of special random impulsive fractional differential equations. Initially, we present the existence of the mild solution via Leray-Schauder fixed point method. After that, we establish the exponential stability of the system. Finally, we give examples to illustrate the effectiveness of the theoretical results.

A mathematical model of an interaction between two populations namely prey and predator is studied based on a Gause-type predator–prey model involving the additive Allee effect and intraspecific competition on the predator. A famous fractional operator called Atangana–Baleanu–Caputo fractional derivative (ABC) is employed to integrate the impact of the memory effect on the dynamical behavior of the model. The existence, uniqueness, non-negativity, and boundedness of the solution are given to confirm the biological feasibility and validity of the model. Three types of equilibrium points are identified on the origin, axial, and interior including their existence conditions. The Lyapunov direct method for the ABC model is used to investigate the asymptotic stability condition for each equilibrium point. The numerical simulations are provided to demonstrate the impact of several biological parameters on the dynamics of the solutions. The emergence of transcritical, saddle–node, and backward bifurcations driven by the Allee constant are provided which resulted in the appearance of bistability condition. A Hopf bifurcation as well as the evolution of the limit-cycle also occurs as the impact of the memory effect. Each analytical and numerical result is biologically interpreted to show the way that the density of both populations always balances in their ecosystem.

The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread.